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Next Generation Math IIITextbookPhilippine Copyright 2011 by DIWA LEARNING SYSTEMS INCAll rights reserved. Printed in the PhilippinesEditorial, design, and layout by University Press of First AsiaNo part of this publication may be reproduced or transmitted in any form or by any means electronic ormechanical, including photocopying, recording, or any information storage and retrieval systems, withoutpermission in writing from the copyright owner. Exclusively distributed by DIWA LEARNING SYSTEMS INC 4/F SEDCCO 1 Bldg. 120 Thailand corner Legazpi Streets Legaspi Village, 1229 Makati City, Philippines Tel. No.: (632) 893-8501 * Fax: (632) 817-8700 ISBN 978-971-46-0184-0AuthorsMaria Maitas M. Marasigan ﬁnished her master’s degree in Mathematics and bachelor’s degree in Mathematicsfor Teachers from De La Salle University–Dasmariñas and Philippine Normal University (PNU), respectively. Sheis a licensed college instructor and has been teaching various Mathematics subjects in several universities and collegesboth in Manila and in Cavite for more than 12 years. She is presently taking up her doctorate degree in MathematicsEducation at PNU and is a full-time academic faculty of Lyceum University of the Philippines–Cavite since June2008.Angelo D. Uy obtained his bachelor’s degree in Mathematics for Teachers at Philippine Normal University, wherehe is also currently pursuing his master’s degree in Education with specialization in Mathematics Education. Hewas a trainer in different Math competitions and a participant in various seminar-workshops sponsored by theMathematics Teachers Association of the Philippines. He taught at the grade school and high school departmentsof Hotchkiss Learning Center in Surigao del Sur and at the grade school department of De La Salle Santiago ZobelSchool in Ayala Alabang. Mr. Uy presently teaches Mathematics at Jacobo Z. Gonzales Memorial National HighSchool (JZGMNHS) in Biñan, Laguna. He is currently the JZGMNHS Math Club adviser and one of the PROs ofthe Secondary Mathematics Teachers Association of Laguna (SEMATAL).Consultant-ReviewerLorelei B. Ladao-Saren obtained her master’s degree in Mathematics, with high distinction, from De La SalleUniversity (DLSU)–Dasmariñas and her bachelor’s degree in Statistics from University of the Philippines–Diliman.She is presently pursuing her doctorate degree in Mathematics Education at Philippine Normal University.Ms. Ladao-Saren was a former director for Research, Publication, and Community Extension Services at World CitiColleges. She has also taught Mathematics at Asia Paciﬁc College, Southville Foreign University, and at DLSU–Dasmariñas. She currently teaches Mathematics at DLSU–College of St. Benilde and at the Graduate School ofRizal Technological University.
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Preface The Next Generation Math series covers topics and competencies that are aligned withthe Basic Education Curriculum (BEC) and the Engineering and Science Education Program(ESEP) of the Department of Education. It is composed of different mathematics disciplines:elementray algebra in ﬁrst year; intermediate algebra in second year; geometry in third year;and advanced algebra, trigonometry, statistics, and calculus in fourth year. It tries to cover themany important topics that will satisfy the needs of different groups of learners. The series supports the constructivist approach to teaching and learning process. Lessonsare presented through meaningful activities which are designed to provide you an opportunityto make different connections between a concrete situations and mathematics. The activitiesare designed to develop your skills in problem solving, critical thinking, decision making, andcreative thinking through exchange of ideas and own discovery. Each book in this series providesopportunities for you to discuss, explore, and construct mathematical ideas and interpret newinformation and knowledge at a different perspective. You will also be able to structure andevaluate your own conjectures and apply previously acquired knowledge and skills. The series has the following salient features:• Lessons are inquiry based, enriched with applicable technologies, and integrated with science and real-life applications.• Emphasis on the development of higher-order thinking skills is evident on the illustrative examples and exercises provided in every lesson. To enhance your mathematics skills, the degree of difﬁculty of the problems ranges from simple to more challenging ones.• Exercises include research work to emphasize the importance of research as a tool in satisfying quest for knowledge and acquiring valuable insights about certain topics.• Historical notes, application of mathematical ideas in future careers, and pieces of trivia are presented in each chapter. It is with a sincere desire to provide a useful tool in enhancing appreciation and betterunderstanding of mathematics that the Next Generation Math series was conceptualized.
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Table of ContentsUnit I The Essentials of GeometryChapter 1 The Art of ReasoningLesson 1 The Reasons Behind ...................................................................................... 2Lesson 2 If-and-Then Statements................................................................................. 8Lesson 3 The Converse, the Inverse, the Contrapositive, and the Biconditional Statements ................................................................ 13Lesson 4 Deductive Reasoning ................................................................................... 18IT Matters ................................................................................................................... 25Chapter 2 The Models of Points, Lines, and PlanesLesson 1 Points, Lines, and Planes in Space .............................................................. 26Lesson 2 Pairs of Lines .............................................................................................. 34Lesson 3 Basic Postulates on Points, Lines, and Planes.............................................. 40Lesson 4 Segment Relationships ................................................................................ 45IT Matters ................................................................................................................... 52Chapter 3 All about AnglesLesson 1 Angles ......................................................................................................... 57Lesson 2 Angle Measurement ..................................................................................... 63Lesson 3 Bisector of an Angle .................................................................................... 70Lesson 4 Pairs of Angles ............................................................................................ 76IT Matters ................................................................................................................... 83Chapter 4 The TransversalLesson 1 Angles Formed by a Transversal Line........................................................... 87Lesson 2 Perpendicular Lines..................................................................................... 94Lesson 3 Parallel Lines ............................................................................................ 100IT Matters ................................................................................................................. 105Chapter 5 The PolygonLesson 1 Polygons.................................................................................................... 109Lesson 2 Angles in a Polygon ................................................................................... 115IT Matters ................................................................................................................. 121
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Unit II The TrianglesChapter 6 The Triangle CongruenceLesson 1 Properties of a Triangle .............................................................................. 126Lesson 2 Congruence on Triangles ........................................................................... 133Lesson 3 Congruent Triangles .................................................................................. 138IT Matters ................................................................................................................. 148Chapter 7 Inequalities in TrianglesLesson 1 Inequalities in a Triangle ........................................................................... 153Lesson 2 Inequalities in Two Triangles ..................................................................... 158IT Matters ................................................................................................................. 164Unit III Quadrilaterals and SimilarityChapter 8 QuadrilateralsLesson 1 Properties of Parallelograms ..................................................................... 166Lesson 2 More on Parallelograms ............................................................................ 172Lesson 3 Properties of Special Quadrilaterals .......................................................... 178Lesson 4 The Trapezoid and Its Properties ............................................................... 185Lesson 5 The Kite and Its Properties ........................................................................ 189Lesson 6 Solving Problems Involving Quadrilaterals ................................................. 193IT Matters ................................................................................................................. 197Chapter 9 SimilarityLesson 1 Proportional Segments .............................................................................. 198Lesson 2 The Basic Proportionality Theorem and Its Converse ................................. 205Lesson 3 Other Proportionality Theorems ................................................................. 211IT Matters ................................................................................................................. 219Chapter 10 More on SimilarityLesson 1 Similar Polygons........................................................................................ 221Lesson 2 Similar Triangles ....................................................................................... 227Lesson 3 Triangle Similarity Theorems ..................................................................... 234Lesson 4 Similarities in a Right Triangle .................................................................. 241Lesson 5 The Pythagorean Theorem and Its Converse .............................................. 248Lesson 6 Special Right Triangles .............................................................................. 254Lesson 7 Areas and Perimeters of Similar Triangles.................................................. 258Lesson 8 Solving Word Problems on Similarity ......................................................... 262IT Matters ................................................................................................................. 268
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Unit IV Circles, Plane and Solid Geometry, and Plane Coordinate GeometryChapter 11 CirclesLesson 1 Parts of a Circle ......................................................................................... 271Lesson 2 Arcs and Central Angles ............................................................................ 276Lesson 3 Inscribed Angles ........................................................................................ 281Lesson 4 Properties of a Line Tangent to a Circle...................................................... 287Lesson 5 Theorems on Chords of a Circle ................................................................. 293Lesson 6 Angles Formed by Tangents and Secants ................................................... 298Lesson 7 Tangent Circles and Common Tangents..................................................... 305IT Matters ................................................................................................................. 312Chapter 12 Plane and Solid GeometryLesson 1 Circumference of a Circle, Perimeter and Area of Plane Figures ................ 313Lesson 2 Surface Area and Volume of Solid Figures ................................................. 319IT Matters ................................................................................................................. 326Chapter 13 Plane Coordinate GeometryLesson 1 The Cartesian Coordinate Plane: A Review ................................................. 328Lesson 2 Equation of a Line and the Point of Intersection of Two Lines..................... 333Lesson 3 Parallel and Perpendicular Lines in the Coordinate Plane .......................... 337Lesson 4 Distance and Midpoint Formulas............................................................... 340Lesson 6 Circles in the Coordinate Plane.................................................................. 346IT Matters ................................................................................................................. 350Glossary ................................................................................................................. 351Bibliography ................................................................................................................. 358Index ................................................................................................................. 360
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The Essentials of Unit IGeometry This unit covers ﬁve chapters that deal with the art of reasoning, points, lines, planes,angles, transversal lines, and polygons. In Chapter 1, you will learn new ways to reason mathematically and to use the differentkinds of reasoning in formulating conjectures. Chapter 2 will help you understand the differentmodels of points, lines, and planes in space, and how they can be used to determine therelationships between segments and to prove theorems. In Chapter 3, you will learn the differentkinds of angles based on their measurements and the different geometric ﬁgures that can bemodeled by the different pairs of angles. Chapter 4 will introduce the angles formed by atransversal line, and the perpendicularity and parallelism of lines. Finally, Chapter 5 will coverconvex and concave polygons and the angles in a polygon. All of these topics are essential toolsin understanding geometry.
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Chapter 1 THE ART OF REASONING Learning Objectives • Deﬁne and differentiate the kinds of reasoning • Deﬁne and apply the different methods of proving • Apply the symbols of logic in solving problems • Use analogies in descriptions, comparisons, and understanding new concepts and ideas • Write a conjecture based on inductive and deductive reasonings • Understand a conditional statement written in an alternate form and rewrite it in the “if-then form” • Write and determine the truth value of the converse, inverse, and contrapositive of a given conditional statement • Determine whether a biconditional statement is true or false • Prove logical arguments deductively • Solve real-life problems where there is possibility of more than one adequate solution • Give reasons for the choices made as well as vary the style and amount of detail in explanations • Demonstrate an advanced ability to read and write different conditional statements into the “if-then” form • Solve problems with an optimistic attitude and open mind Lesson 1 The Reasons Behind Power Up1. Look closely at each ﬁgure. Which is longer, P or X? X P2 Next Generation Math III
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2. What conjecture can you make out of this statement? Point is to segment as arrow is to _______.3. What would be the next set of numbers? 1 1 2 2 1 1 3 4 3 1 1 4 7 7 4 1 How did you get your answer?4. Consider the following statements: A student likes geometry. Maria is a student. Are the two statements related? If so, how? What made you say so? What can you conclude from the two statements? Walk Through• Statement – expression with a complete thought• Hypothesis – statement where a conclusion is being drawn• Conclusion – relation that can be drawn from the given statements• Conjecture – unproven statement or proposition that is based on observations• Reasoning – process of determining the relation between the given hypothesesKinds of Reasoning• Inductive reasoning – reasoning that is based on observing and recognizing patterns in a set of data and using the patterns to arrive at a conjecture• Deductive reasoning – reasoning that uses facts, rules, deﬁnitions, or properties in a logical order to show that a desired conclusion is trueExample 1: Determine the kind of reasoning used in each statement below. a. Given: • • • ? Thus, the next ﬁgure is • . • • • • • • • • • • • • • • • • b. A line has points. A triangle has lines. A triangle has points. EXTEND THE CONCEPTSolution: Scientiﬁc Method a. Inductive reasoning The scientiﬁc method uses inductive reasoning b. Deductive reasoning and deductive reasoning. It consists of deﬁning the problem, making an educated guess or hypothesis, gathering facts from observations and experiments, classifying data, logically drawing conclusions, and proving the hypothesis to reach conclusions. The Essentials of Geometry 3
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Example 2: Identify the hypothesis and the conclusion of each statement. a. Given: Thus, b. Given: x – 2 = 6. Thus, x = 8. c. A plane is a closed ﬁgure. A square is a plane. A square is a closed ﬁgure. d. Given the ﬁrst ﬁve terms of the sequence 3, 5, 7, 9, 11. Then, the sixth term is 13.Solution: a. Hypothesis: Conclusion: b. Hypothesis: x – 2 = 6 Conclusion: x = 8 c. Hypotheses: A plane is a closed ﬁgure. A square is a plane. Conclusion: A square is a closed ﬁgure. d. Hypothesis: 3, 5, 7, 9, 11 Conclusion: 13 is the sixth termExample 3: Supply a word or a number that will make each statement true. a. Circumference is to circle as _____ is to square. b. _____ is to angle as meter is to length. c. 2, 4, 5, 10, 11, __, 23 d. 9:3 = __:5 e. If x – 5 = 6, then x = ___.Solution: a. perimeter b. Degree or Radian c. 22 d. 15 e. 11Example 4: Make a conjecture based on inductive reasoning. Justify your conclusion. a. b. c. 1, 2, 4, 8, 16 d. 1, 2, 6, 24, 1204 Next Generation Math III
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Solution: a. The number of dots in each row and column is one more than the number of dots in the previous row and column. b. The dot inside the diamond rotates counterclockwise, while the dot inside the rectangle rotates clockwise. c. 32. Each term after the ﬁrst is obtained by doubling the previous term. d. 720. Each term is obtained by applying the following solution: 1 × 2 = 2, 2 × 3 = 6, 6 × 4 = 24, 24 × 5 = 120, 120 × 6 = 720Example 5: Write a conclusion using deductive reasoning. a. All students who passed algebra are enrolled in geometry class. Carlo passed algebra. b. The number of vertices in a polygon also determines the number of sides. A square has 4 vertices. c. Lines that meet at a common point are intersecting lines. Line m and line n meet at a common point. d. An acute angle measures less than 90. ∠K is an acute angle. Note: In this book, we shall consider angle measures in degrees.Solution: a. Carlo is enrolled in geometry class. b. A square has 4 sides. c. Line m and line n are intersecting lines. d. ∠K measures less than 90. Move UpI. Identify whether the conclusion was a result of induction or deduction. _____________ 1. A quadrilateral has four sides. A rectangle has four sides. A rectangle is a quadrilateral. _____________ 2. If an angle is obtuse, then it cannot be acute. ∠M is obtuse. ∠M cannot be acute. _____________ 3. If 3, 6, 9, and 12 are the ﬁrst four terms, then 15 is the ﬁfth term. _____________ 4. Ray is to an angle as side is to a triangle. _____________ 5. Trees can be found in a forest. Birds live on trees. Therefore, birds can be found in a forest. _____________ 6. If 3:6, 6:18, and 12:54, then 24:162. The Essentials of Geometry 5
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_____________ 7. All animals are mortal. All humans are animals. Therefore, all humans are mortal. _____________ 8. If a polygon has ﬁve sides, then it is a pentagon. 9. All trees are plants. _____________ All acacia are trees. Thus, all acacia are plants. _____________ 10. If a number is divisible by 2, then it is even.II. Encircle the letter that corresponds to the correct answer. 1. Which of the following is not a process? a. statement b. analogy c. deduction d. induction 2. Which reasoning is based on facts and not on patterns? a. deduction c. analogy b. induction d. none of these 3. Which of the following ﬁgures have the same length? i. ii. iii. iv. a. i and ii b. ii and iv c. i and iii d. ii and iii For numbers 4 and 5, consider the statement: If two points determine a line, then three distinct points determine a plane. 4. Which of the following is the hypothesis? a. Three points determine a line. c. Three points determine a plane. b. Three distinct points determine d. Two points determine a line. a plane. 5. Which of the following is the conclusion? a. Three points determine a line. c. Two points determine a plane. b. Three distinct points determine d. Two points determine a line. a plane. For numbers 6 and 7, pick another pair that has the same relationship as the given pair. 6. 3:9 as _________. a. 9:3 b. 3:1 c. 12:36 d. 12:60 7. Thermometer is to temperature as _________ is to _________. a. telescope, astronomy c. clock, minutes b. scale, weight d. microscope, biologist6 Next Generation Math III
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8. Determine the next ﬁgure in the given set of ﬁgures below. a. b. c. d.For numbers 9 and 10, consider the two statements below. The number 11 is a prime number. A prime number has factors of 1 and itself. 9. What conclusion can you draw out of the given statements? a. The number 11 is a prime number. b. The number 11 is factorable. c. The number 11 has factors of 1 and itself. d. The number 11 is a factor. 10. What kind of reasoning should be used to draw the conclusion? a. deduction c. analogy b. induction d. none of theseIII. Read and analyze each problem carefully. 1. What can you conclude about the size range of carrots if you measure 20 carrots and found that they are all between 6 and 8 inches long? Which type of reasoning are you going to apply? Justify your answer. 2. As you enter a room, you suddenly observe that the room is dark. You wonder why the room is dark and you attempt to ﬁnd explanations out of curiosity. You thought of a lot of possibilities why the room is dark. You might think that the lights are turned off or the room’s light bulb has burnt out, and worst, you could be going blind. What are you going to do to test your hypotheses? Will you apply inductive or deductive reasoning? Justify your answer. 3. Look for an article that shows how deductive and inductive reasonings are used in scientiﬁc studies. Share it with the class and let the class formulate conjectures from the article. Cite your references. (For example: If 300 out of 1 000 people got better in taking vitamins, then 30% of any population might also get better in taking vitamins.) The Essentials of Geometry 7
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